The present invention relates to phase sensitive demodulators and, more particularly, to a harmonic insensitive phase sensitive demodulator. To put the present invention into perspective, it is useful to consider the operation and limitations of conventional full-wave phase sensitive demodulators and analog multipliers.
A conventional full-wave phase sensitive demodulator employs a reference Er and a signal Es as inputs, and produces an output having an average value which is proportional to the cosine of the phase angle between the two inputs. The operation of the full-wave phase sensitive demodulator may be better understood by considering it to involve the multiplication of the signal Es by a square wave Er having an amplitude which is alternately +1 and -1. The square wave may be expanded in a Fourier series as: EQU Er = (4/.pi.) (sin wt + 1/3 sin 3 wt + 1/5 sin 5 wt + . . .) (1)
By use of the following trigonometric identities: EQU sin w.sub.1 t .multidot. sin w.sub.1 t = sin.sup.2 w.sub.1 t = 1/2- 1/2 cos 2w.sub.1 t (2) EQU sin w.sub.1 t cos w.sub.2 t = 1/2 sin (w.sub.1 +w.sub.2)t + 1/2 sin (w.sub.1 - w.sub.2)t sin w.sub.1 t cos w.sub.1 t = 1/2 sin 2w.sub.1 t (3)
it is evident that a DC output will result only if the signal Es contains frequencies in the reference square wave which are not in quadrature phase with the reference frequency components.
All other frequencies and quadrature components of the reference frequencies will result in only AC components called ripple. In short, a full-wave phase sensitive demodulator will exhibit response to a fundamental and to odd order harmonics in inverse proportion to the harmonic order. This response to harmonics is generally undesirable and it is an object of this invention to produce a phase sensitive demodulator that will eliminate as many of the odd order responses as desired, while maintaining the switching nature of the demodulator.
It is evident that, in the description of the full-wave phase sensitive demodulator, if Er were a sinusoid of a given frequency, an output DC would result only if that frequency were present in the signal Es. This could be done by using a linear analog multiplier. This technique has the undesirable characteristic of producing a DC output proportional to both the signal and reference, requiring some means for accurately controlling the reference amplitude. Analog multipliers also have poor temperature stability and non-linearities.
The prior art has managed to render the outputs of phase sensitive demodulators, also known as phase sensitive detectors, insensitive to the presence of certain odd harmonics of the reference frequency which are present in the input signal.
Thus, in U.S. Pat. No. 3,839,716 there is taught the use of ".pi./6" chopping waveforms which do not contain the third, ninth, fifteenth, twenty-first, etc. odd harmonics. This results in a demodulator output which is not sensitive to the presence of those odd harmonics in the input signal. In U.S. Pat. No. 3,517,298 a technique is disclosed for eliminating certain pairs of odd harmonics, e.g., the 3rd and 5th; the 11th and 13th; the 19th and 21st; the 27th and 29th; etc. Thus, while the prior art teaches the elimination of certain odd harmonics, it does not teach that one may, to any desired degree, eliminate the effects of harmonic distortion. That is the subject of the instant invention.
As can be seen from the previous discussion, if the multiplying signal Er were perfectly sinusoidal, the resultant multiplication would yield a DC output proportional only to the component of the signal Es that is the same frequency as the reference [Equation (2)] and thus be independent of harmonics. Using digital techniques it is possible to synthesize a sinusoid of a purity limited only by the digital resolution. This can be done by considering the synthesized waveform as analogous to that of a sinusoid which has been passed through a sample and hold system. Sampled data theory indicates that the frequency content of this waveform is w, fs .+-. w, 2fs .+-. w, 3fs .+-. w, etc. Where the sampling frequency fs is equal to 16w, the synthesized waveform has the following Fourier series: EQU sin wt + 1/15 sin 15wt + 1/17 sin 17wt + 1/31 sin 31wt + 1/33 sin 33wt +
As is apparent, the 3rd, 5th, 7th, 9th, 11th, and 13th harmonics have been eliminated, thereby eliminating any DC outputs due to their presence in the signal.
If one wished to eliminate the 15th and 17th harmonics as well one could synthesize a sinusoidal waveform having the following Fourier series by using a sampling frequency fs of 20w. EQU sin wt + 1/19 sin 19wt + 1/21 sin 21wt + 1/39 sin 39wt + 1/41 sin 41wt + 1/59 sin 59wt + 1/61 sin 61wt +
In general, to render the demodulator output insensitive to the presence of the nth and lower odd harmonics of the reference frequency in the input signal one should synthesize a sinusoidal waveform having the following Fourier series by using a sampling frequency of (n+3)w or a multiple thereof: sin wt + 1/n+2 sin (n + 2)wt + 1/n+4 sin (n +4)wt + 1/2n+5 sin (2n+5)wt + 1/2n+7 sin (2n+7)wt + 1/3n+8 sin (3n+8)wt + 1/3n+10 sin (3n+10)wt + In practice, n should have a value of at least 7 and preferably as high as 13 or 29 for high levels of harmonic rejection. In practice, rejecting the 29th and lower harmonics has been found to be particularly efficacious.
As will be appreciated, because of limitations imposed by tolerances, digital resolution, etc. it is not practical to synthesize a function having exactly the above Fourier series. It is sufficient, however, if one synthesizes a function having substantially that Fourier series, which will ensure that the demodulator output is rendered substantially insensitive to the nth and lower odd harmonics of the reference frequency which are present in the input signal Es. Obviously, the more closely the Fourier series of the synthesized function approaches the ideal the more complete will be the harmonic rejection.